Find each quotient. Write answers in standard form. 2-i / 2+i

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Identify the given expression: \(\frac{2 - i}{2 + i}\), where \(i\) is the imaginary unit with \(i^2 = -1\).

To simplify the quotient, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 + i\) is \(2 - i\). So multiply by \(\frac{2 - i}{2 - i}\).

Perform the multiplication in the numerator: \((2 - i)(2 - i)\). Use the distributive property (FOIL) to expand this product.

Perform the multiplication in the denominator: \((2 + i)(2 - i)\). This is a difference of squares, so use the formula \(a^2 - b^2\) where \(a=2\) and \(b=i\).

Simplify both numerator and denominator by combining like terms and using \(i^2 = -1\). Then write the resulting expression in the form \(a + bi\), which is the standard form for complex numbers.

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주요 개념

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Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. Writing answers in standard form means presenting the result explicitly as a sum of a real number and an imaginary number.

Division of Complex Numbers

Dividing complex numbers involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator. This process simplifies the expression into standard form.

Complex Conjugates

The complex conjugate of a number a + bi is a - bi. Multiplying a complex number by its conjugate results in a real number, which is useful for rationalizing denominators in division problems.